Transcript A Brief Introduction to Quantum Computation 1

A Brief Introduction to
Quantum Computation1
Melanie Mitchell
Portland State University
1
This talk is based on the following paper:
E. Rieffel & W. Polak,
An introduction to Quantum Computing for Non-Physicists.
ACM Computing Surveys, 32(3) 300-335, 2000.
History of Quantum Computation
1982: Richard Feynman, “Simulating physics with
computers”
• Notes that simulating quantum systems with classical
computers is exponential in the number n of particles in the
system
• Suggests that quantum effects could themselves be used to
make such simulations polynomial in n
• General idea: due to “superposition” and “entanglement”,
state space encoded by quantum system of n particles is
exponentially larger than state space encoded by classical
system of n particles.
• Could this exponential increase in state space be harnessed
to create exponential parallelism (2n) with n computing
elements?
• At that time, no one had any workable ideas about how to
build a computer that exploits quantum mechanics.
• Also, no one had a detailed model of what a quantum
computer would be like.
• Also, no one had any algorithms for actually computing
anything on a quantum computer.
• 1985: David Deutsch, “Quantum theory, the ChurchTuring principle and the universal quantum computer”
– Developed the notion of a universal quantum computer
– Gave a simple example of a quantum algorithm that is
faster than any known classical algorithm for the same
problem
• 1994, Peter Shor, “Polynomial time algorithms for prime
factorization and discrete logarithms on a quantum
computer”
– Gave quantum algorithm for factoring large numbers in
polynomial time, and for performing discrete logs
– First major algorithmic result in field of quantum
computation
– Got a lot of people interested in the field
– DARPA, NSF, others started throwing money at it
Other results:
• Cryptography: Secure private key communication using
qubits.
• Grover’s search algorithm: Searches an unstructured list
of n items in O(sqrt(n)) on a quantum computer. Known
algorithms for classical computers are no better than O(n).
• Theory of quantum error correction
“It is as yet unknown whether the power of quantum
parallelism can be harnessed for a wide variety of
applications.”
Open question: Can any NP-complete problem be solved
in polynomial time on a quantum computer?
Polarization experiment
• Polarization of light: electric field vector is confined to a
single direction. This is a quantum-mechanical property of
photons.
• Light source: Assume each photon has a random
polarization
• Filter A: Horizontally polarized
A
photons randomly
polarized
all photons
horizontally
polarized
1/2 original intensity
Polarization experiment
• Polarization of light: electric field vector is confined to a
single direction. This is a quantum-mechanical property of
photons.
• Light source: Assume each photon has a random
polarization
• Filter A: Horizontally polarized
• Filter B: Vertically polarized
A
photons randomly
polarized
B
all photons
horizontally
polarized
no light
Polarization experiment
• Polarization of light: electric field vector is confined to a
single direction. This is a quantum-mechanical property of
photons.
• Light source: Assume each photon has a random
polarization
• Filter A: Horizontally polarized
• Filter B: Vertically polarized
• Filter C: polarized at 45 degrees
A
C
B
photons randomly
polarized
1/8 original intensity
Polarization experiment explanation
• Photon’s polarization state can be modeled by a unit vector
pointing in the appropriate direction
• Let the basis vectors be  and 
• Any polarization can be expressed as
  a    b 
where a and b are complex numbers such that
a  b 1
2
2
• Measurement postulate of quantum mechanics
– Any device measuring a quantum system has an
associated orthonormal basis with respect to which the
measurement takes place.
– Measurement of a state transforms the state into one of
these basis vectors.
– The probability that the state is measured as basis
vector u is the norm of the amplitude of the component
of the original state in the direction of u.
• Example:
Let   a  b be the polarization state of a photon.
Then this state will be measured as  with probability |a|2
and as  with probability |b|2 .
These are indeed probabilities, since a  b  1 .
2
2
The measurement will also change the state to the result
of the measurement.
Its original value cannot be determined.
• Back to the polarization experiment:
– Original light source produces photons with random
polarizations:
  a   b
– Thus 50% are measured as horizontal, and let through.
(Why?)
– Those photons are now in the horizontal state,
None are let through by the vertical filter.
• Now let the 45o filter be put behind the horizontal filter.
The 45o filter measures photon polarization with respect to
a new basis:
• Photons in state  will be measured as
probability 1/2. So half will get through.
with
• Likewise, filter B will let half of these through.
• Total photons getting through all three filters: (1/2)3 = 1/8.
Bra/Ket notation
Ket:
x
• Column vector, used to describe quantum states
• E.g.,
 0,1 
can be written as
 (1, 0)
• In general,
x  a 0  b 1  (a, b)T
Bra: x
• Conjugate transpose of
x  (a , b )
x
T
, (0,1)T

Quantum bits (“qubits”)
A qubit is a unit vector in a two-dimensional complex vector
space.
Assume a particular basis, denoted by
0,1 
These basis states represent the classical bit values 0 and 1.
However, unlike classical bits, qubits can be in a
superposition of these two states, e.g.,
a 0 b1
such that
a  b 1
2
2
Suppose a qubit (a, b)T is measured with respect to the basis
 0,1 
Probability that 0 will be measured is |a|2, and
Probability that 1 will be measured is |b|2.
When a qubit is measured, its state becomes the basis state
that was measured. All information about the original
superposition is lost.
Example of using qubits:
Quantum key distribution
(Bennett & Brassard, 1987)
•
Solves problem of secure key distribution for private key
cryptography
•
Public key cryptography using RSA is great, but relies on
factoring being a hard problem.
•
Private key cryptography is in principle more secure, but
relies on private key being securely transmitted.
•
Quantum method gives perfect security in private key
transmission
Alice wants to send a key to Bob by encoding each bit as
the quantum state of a photon.
She secretly encodes each bit by randomly choosing one of
two basis:

0
1
or
0
1
E.g.,
1
1
0
1
1 1 0 0 0 0
Bob measures the state of each photon he receives by
randomly picking either basis, up-down (ud), or diagonal
(diag).
What Alice sent
1
1
0
1
1 1 0 0 0 0
What Bob measured
Basis:
Measurement:
ud
1
diag ud diag ud ud diag ud diag diag
0
0
1
1 1
1 0 1 0
Now Alice and Bob communicate what bases they used for
each measurement. They save only the bits for which they
used the same bases (approx. 50%). These bits form the
private key.
Bob measures the state of each photon he receives by
randomly picking either basis, up-down (ud), or diagonal
(diag).
What Alice sent
1
1
0
1
1 1 0 0 0 0
What Bob measured
Basis:
Measurement:
ud
1
diag ud diag ud ud diag ud diag diag
0
0
1
1 1
1 0 1 0
Now Alice and Bob communicate what bases they used for
each measurement. They save only the bits for which they
used the same bases (approx. 50%). These bits form the
private key.
Now, suppose a third person, Eve, intercepts the stream of
photons from Alice, measures their states, and sends new
photons with the measured states to Bob.
She will choose the wrong basis 50% of the time.
So on the photons for which Bob chooses the same basis as
Alice, 25% of them will be incorrect.
Bob and Alice can detect this error by sending sample
“correct” (parity) bits to each other over open
communication line. Thus they can detect an
eavesdropper.
Multiple qubits
• Suppose you have n particles whose states are each vectors
in a 2-D vector space (e.g., x, y position)
• How many dimensions do you need to describe complete
state of the system?
– Particles have coordinates (x1, y1), (x2, y2), ..., (xn, yn)
– Complete state is simply a list of all the coordinates:
(x1, y1, x2, y2, ..., xn, yn)
– I.e., 2n dimensions
• Quantum case is different due to “entanglement”.
• Again, suppose you have n particles whose states are each
vectors in a 2-D vector space.
– Particles have coordinates (x1, y1), (x2, y2), ..., (xn, yn)
– Complete state is more complicated, due to
entanglement of states:
(x1x2x3...xn-1xn, x1x2x3...xn-1yn, x1x2x3...yn-1xn, ...)
– I.e., 2n dimensions
y
(1, 2)
Example: n = 2
(2, 1)
– Classical basis vectors: (1, 0), (0, 1)
– Classical state of system: ( (1, 2), (2, 1) )
x
– Quantum basis vectors:
(1, 0)  (1,0) = (1, 0, 0, 0)
(1, 0)  (0,1) = (0, 0, 1, 0)
(0, 1)  (1,0) = (0, 1, 0, 0)
(0, 1)  (0, 1)= (0, 0, 0, 1)
– Quantum state of system: Linear combination of those
basis vectors
y
(1, 2)
(2, 1)
x
Interpretation: In quantum system you now need four
numbers to give the state of the two-particle system.
Each dimension describes a particular entanglement of
the original dimensions of the individual particles.
In quantum computation, the “particles” are the qubits.
Each qubit has a state defined by two complex numbers:
  a 0  b 1 , where
 1
 0
0    and 1   
 0
 1
A two-qubit state is defined by four complex numbers:
q  a 00  b 01  c 10  d 11 , where
 1
 0
 0
 0
 
 
 
 
 0
 0
 1
 0
00    , 01    , 10    , 11   
0
1
0
0
 
 
 
 
 0
 0
 0
 1
 
 
 
 
• Recall the notion of superposition:
– A single qubit is in a superposition of two basis states:
  a 0 b1
– A 3-qubit system is in a superposition of eight basis
states:
  a1 000  a2 001  a3 010  a4 011  a5 100  a6 101  a7 110  a8 111
where a1, a2, ..., an are complex numbers such that
a1  a2  a3  a4  a5  a6  a7  a8  1
2
2
2
2
2
2
2
2
• In general, an n-qubit system is in a superposition of 2n
basis states.
• This exponential increase in size of state space as n
increases is why classical computers cannot efficiently
simulate quantum systems.
• This was Feynman’s impetus for proposing quantum
computers.
Measurement in entangled systems
• Example 1: The state
1
( 00  11 )
2
is entangled:
– If no bits have yet been measured, the probability of
measuring the first (or second) bit as 0 is 0.5.
– However, if the second (or first) bit has been measured,
the probability is 0 or 1, depending on what the second
(or first) bit was measured to be.
1
( 00  01 ) is not entangled:
• Example 2: The state
2
Can factor out first bit: it will always be measured as 0 .
Measuring it does not affect the second bit.
EPR paradox
(Einstein, Podolsky, Rosen, “Can quantum-mechanical description of
physical reality be considered complete?” (1935))
• Imagine a source that generates two maximally entangled
1
particles:
( 00  11 )
2
• One particle is sent to Alice, and the other to Bob.
• Suppose Alice measures her particle and observes state 0 .
• Then the combined state will now be 00 . This means
Bob will also measure 0 .
• The change in state occurs instantaneously, no matter how
far away Alice and Bob are from each other.
• Does this allow communication faster than the speed of
light?
• EPR’s explanation:
– Each particle has a “hidden” internal state that determines what the
result of any given measurement will be.
– Called “local hidden variable theory”.
– Einstein: “God does not play dice.”
• Problem with this explanation:
– Doesn’t explain measurements with respect to a different basis.
– Bell’s theorem: predicts an inequality that must be satisfied if
hidden variable theory is correct.
– Experiments show it to be incorrect.
• Cause and effect explanation:
– Somehow Alice’s measurement causes the result found by Bob,
• Problems with this explanation:
– Violates relativity theory: Can set up experiment where one
observer sees Alice do measurement first, while other observer
sees Bob do measurement first. This does not change the results
of the measurements.
– This also shows that Alice and Bob can’t use their particles to
communicate faster than the speed of light.
– “All that can be said is that Alice and Bob will observe the same
random behavior.”
• From Black et al., “Quantum computing and
communications” (2002):
“The ugly truth is that general relativity and quantum
mechanics are not consistent. That is, our current
formulations of general relativity and quantum
mechanics give different predictions for extreme cases.
We assume there is a ‘Theory of Everything’ that
reconciles the two, but it is still very much an area of
thought and research.
Since relatively is not needed in quantum computing,
we ignore this problem.”
The four basic single-qubit transformations
I:
0  0
1 1
 1 0


 0 1
X:
0 1
1  0
 0 1


 1 0
Y:
Z:
0 1
1  0
 0 1


 1 0
0  0
1 1
1 0 


 0  1
Hadamard transformation
H:
1
0 
(0 1)
2
1
1 
(0 1)
2
1 1 1 


2 1  1
How are I, X, Y, Z, and H implemented in a physical
quantum computer?
That’s beyond the scope of this talk....
Walsh-Hardamard Transformation
W : ( H  H    H ) 00  0

1
2

n
2 n 1
1
2
( ( 0  1 )  ( 0  1 )  ( 0  1 ) )
n

x
x0
For example:
( H  H  H ) 000


1
2
1
3
2
3
(( 0  1 ) ( 0  1 ) ( 0  1 ) )
( 000  001  010  011
 100  101  110  111
Quantum Gate Arrays
• Uf is a quantum gate array that implements function f
(Deutsch proved Uf can be constructed for any computable
function f)
• To compute f(x), apply Uf to x tensored with a “register”
of 0 s :
U f x,0  x, f ( x)
Quantum Parallelism
•
To get exponential parallelism, apply Uf to an input that is
in a superposition of all inputs.
•
Result will be superposition of all results of applying f(x)
to all inputs.
•
Quantum algorithm to compute function f(x):
1. Start with n-qubit state 00...0
2. Apply Walsh-Hadamard transformation W to this state to
get a superposition of all integers in [0, 2n-1]:
( H  H    H ) 00 0

1
2

n
2 n 1
1
2
( 00...0  00...1  ...  11...1 )
n
x
x 0
3. Tensor this state with a “register” of one or more zero
qubits:
2 n 1
1
2
n
 x ,0
x 0
4. Now apply Uf to this state:
 1
U f 
n
2

2 n 1

x 0
  1
x,0   
n
2
 
  1
U f x,0   

n
x 0
2
 
2 n 1
2 n 1

x 0

x, f ( x) 

5. Result is superposition of f(x) for all integers x[0,2n-1]
6. This is what is meant by quantum parallelism.
•
Example: Consider quantum AND gate (“Toffoli gate”):
T : x, y ,0  x, y, x  y
•
Now prepare superposition of all possible bit
combinations of x and y, together with the “register” 0 :
1
( 000  010  100  110 )
2
•
Now, apply T to this superposition:
1

T   000  010  100  110 
2

1
  000  010  100  111 
2
•
The resulting quantum state, obtained in one step, gives a
superposition of all possible results for the AND function
for two bits.
•
What happens if we measure this quantum state?
•
Need clever techniques to allow user to read out a
desired result.
Photo gallery
Peter Shor
Edward Fredkin
Tomosso Toffoli
David Deutsch
Richard Jozsa
Very brief outline of Shor’s algorithm for
factoring
We have a composite number M, and we want to find a factor
of M.
Approach: Reduce the problem of factoring to the
problem of finding the period of an integer function,
f(x)=ax (mod M), for some a.
Problem there is no known classical polynomial time
algorithm to find period r.
1. Guess integers a and m (with some constraints). Define
f(x) = ax (mod M).
2. Create a quantum state of m zero qubits along with a extra
register of qubits: 00...0,0
Apply the Walsh-Hadamard transform to obtain an equalprobability superposition of all integers from 0 to 2m-1:
2 m 1
1
2
m

x0
x,0
3. Now use Uf to compute f (x) for all integers from 0 to
2m-1 simultaneously:
2 m 1
1
2
m

x, f ( x )
x0
4. From this state, create a new (related) state whose
amplitude has the same period as f.
To do this, measure the qubits that encode f(x). Suppose
it is measured as u.
Actually, we don’t care about u; we only care about the
side effect of measurement: the quantum state is now
2 m 1
C  g ( x ) x, u
x0
where C is a scale factor, and
 1 if f ( x)  u
g ( x)  
0 otherwise.
Note that the period of g(x) equals the period of f (x).
If we could measure two successive x’s in the sum, we
would know the period. But we can only do one
measurement.
5. Take the quantum Fourier transform of this new state.
6. Extract the period r. If period is a power of 2, this can be
done exactly. If not, “guess” the period, using “continued
fractions”.
7. Now can apply Euclid’s algorithm (polynomial time) to
obtain a factor of M from the period r. If it fails, start
over with a new guess for a and m. Shor showed that this
algorithm has high probability of succeeding after only a
few repetitions.